In a display of sympathy and solidarity with the Marathon bombing victims, charitable donations are pouring in from across the country and beyond.

Within 24 hours of Monday’s tragedy, The One Fund, the central charity quickly set up for those affected by the bombings, had received $7.5 million. Much of that came from corporate and ­major individual donors, but more than 8,500 ­individuals also contributed to the website, ­onefundboston.org, city officials said.

Internet fund-raising campaigns for those wounded in the bombings have also drawn waves of support, with one raising more than $200,000 in a single day.

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A support fund set up by a high-tech nonprofit group just hours after the attack had raised $130,000, far more than expected. Boston hospitals and charities have been inundated with offers of assistance.

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“It’s been a huge outpouring of people wanting to help,” said Ashley Studley, a spokeswoman for the American Red Cross of Eastern Massachusetts. “It’s incredible. I think people really feel inspired to do something.”

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The Red Cross, which said it has sufficient blood supplies, has directed donors to The One Fund, which was established by state and city officials Tuesday. The campaign was launched with a $1 million donation from insurance giant John Hancock.

Millions have poured in since, and Mayor Thomas M. Menino said Wednesday that he was moved by the magnitude of the response.

“It’s so heartening to me,” he said. “I would never have ­expected anything of this scope. People want to be part of this, to make sure Boston keeps moving forward.”

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Kenneth Feinberg, the lawyer who oversaw compensation of Sept. 11 victims and other similar relief efforts, will design and administer the fund, which is meant to help families most affected by the bombing.

“I want a proven administrator with this,” Menino said.

Feinberg, who will arrive in Boston Friday to begin work on the fund, will work pro bono, Menino said.

In a statement, Feinberg said he was honored to serve at the request of ­Menino and Governor Deval Patrick.

“I will do my best to justify their confidence in me as we move forward to design and ­administer an effective program following the terrible tragedy in Boston,” he said.

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On a smaller scale, other fund-raising efforts have ­exceeded the most optimistic expectations. David Brown, who directs a nonprofit group, Technology Underwriting Greater Good, said he initially hoped to raise $50,000 over the next two to three weeks. Within 36 hours, donations had topped $130,000.

Donations ranged from $5 to $5,000, with an average of $50. Organizers are in touch with city officials, hospitals ­caring for victims, and other ­responders to determine how the money can best be spent.

Grass-roots appeals were ­also drawing strong support. An Internet campaign to support Celeste and Sydney ­Corcoran, a mother and daughter badly injured in the attack, raised more than $230,000 in one day, with more than 3,450 donations.

“We are in complete and ­utter awe of everyone’s generosity,” read the site. “Your support will make such a difference in their lives. Thank you again and again.”

At local hospitals, offers of assistance came in from near and far.

“The good that people are showing, it’s inspiring,” said Beth Donegan Driscoll, who ­directs Child Life Services at Boston Children’s Hospital. “It makes you believe again.”

Some people offered to host the families of patients, others wanted to donate to the hospital. One company offered the use of its private plane.

At Massachusetts General Hospital, some donors sought to support charities that ­marathoners had embraced. Others gave to the hospital’s emergency or social services programs.

“Despite the darkness we’ve gone through, you’re seeing a tremendous amount of light,” said Jim Thompson, the hospital’s vice president for development.

Amid the generosity, officials warned of the potential for fraud, noting that more than 100 Web addresses with ­Marathon-related names, many likely masquerading as charities, have been registered since Monday. Officials at the attorney general’s office and the ­Better Business Bureau warned people to do research before ­donating.

“Whenever there’s a natural disaster or a tragedy, these scam artists will try to take ­advantage of people and their good will,” said Paula Fleming of the Better Business Bureau. “It’s just horrific.”

The bureau had received ­numerous calls from people looking for advice, and steered callers to The One Fund.

At Children’s Hospital, the calls haven’t stopped since Tuesday morning: Stuffed animals for children, gift cards to CVS and restaurants, food baskets for the staff. Pizza for the entire emergency room staff on Tuesday, courtesy of a donor in Georgia.

At a time of grief and deep uncertainty, that goes a long way, Driscoll said.